This paper presents the investigation on the saddle-node bifurcation characteristics of an asymmetrical Duffing system with constant excitation. The Harmonic Balance method is used to obtain the periodic solutions of the system under primary resonance. The Floquet theory is used to analyze the stabilities of the obtained periodic solutions. According to the special geometric feature that the amplitude-frequency curve has the vertical tangent line at the saddle-node bifurcation point，the saddle-node bifurcation sets of the system are calculated. In addition，the influence of the system parameters such as the damping and the magnitude of the harmonic excitation on the saddle-node bifurcation sets are studied. The results show that there are two curves of saddle-node bifurcation sets on the parameter plane of the value of constant excitation and the frequency of the harmonic excitation，one of which is corresponded to the resonance hysteresis with softening characteristics，the other is corresponded to the resonance hysteresis with hardening characteristics. The parameter regions inside the two curves have multiple solutions. Specifically，in the overlapping area of the two multiple solution regions，there are five solutions co-existing and complex vibration jumping phenomenon in the system. With the increase of the constant excitation，the softening characteristic becomes stronger，while the hardening characteristic becomes weaker，the corresponding two resonance hysteresis regions change from being separated to being crossing until the reso? nance hysteresis region with hardening characteristics disappears. Moreover，the multiple solutions co-existing and complex vibration jumping phenomenon can be suppressed by increasing the damping or decreasing the magnitude of the harmonic excitation.